Reduced group C*-algebra - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group and let $\lambda:G\to\mathcal U(L^2(G))$ be its left regular representation. Integrating $\lambda$ gives a $*$ -representation of the group convolution algebra $L^1(G)$ on $L^2(G)$ . The reduced group $C^*$ -algebra is

C_r^*(G)=\overline{\lambda(L^1(G))}^{\,\|\cdot\|} \subseteq\mathcal B(L^2(G)).

Equivalently, it is the completion of $L^1(G)$ after quotienting by the kernel of $\lambda$ and using the reduced norm $\|f\|_r=\|\lambda(f)\|$ . The construction therefore records exactly the part of the group convolution algebra visible in the regular representation.

Discrete groups

If $G$ is discrete, $L^2(G)=\ell^2(G)$ and

\lambda_s\delta_t=\delta_{st}.

Then $C_r^*(G)$ is the norm closure of the finite sums $\sum_{s\in G}a_s\lambda_s$ . It is unital, with unit $\lambda_e$ . For a nondiscrete locally compact group the Dirac mass at $e$ does not lie in $L^1(G)$ , and the reduced group $C^*$ -algebra is generally nonunital; in fact it is unital exactly when $G$ is discrete.

Relation to the full completion

The reduced norm is bounded above by the universal group $C^*$ -norm, so there is a canonical surjective $*$ -homomorphism from $C^*(G)$ onto the reduced completion:

C^*(G)\longrightarrow C_r^*(G).

This map need not be injective. Its being an isomorphism is equivalent to amenability of $G$ , a fact that separates regular-representation data from the totality of unitary-representation data.

Abelian and geometric perspectives

For a locally compact abelian group, the Fourier transform identifies $C_r^*(G)$ with $C_0(\widehat G)$ , where $\widehat G$ is the Pontryagin dual, and the full and reduced completions coincide. For general groups, properties of $C_r^*(G)$ —such as simplicity, exactness, and the existence of traces—encode rigidity, approximation, and dynamical features of $G$ . These properties are not determined merely by the abstract vector space $L^1(G)$ ; the reduced operator norm is essential.

References

Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, American Mathematical Society, 2008. DOI record. Relevant: Chapter 2 and Appendix D on reduced group $C^*$ -algebras and amenability.
Dana P. Williams, Crossed Products of C-Algebras*, American Mathematical Society, 2007. DOI record. Relevant: §2 on full and reduced group $C^*$ -algebras.